×

A reproducing kernel method for nonlinear C-q-fractional IVPs. (English) Zbl 1496.65084

Summary: Here a scheme for solving the Caputo type q-fractional (C-q-fractional) initial value problems (IVPs) in reproducing kernel spaces is given. By the attribute of the q-fractional operator, we first convert the q-fractional differential problems into q-fractional Volterra integral problems. Then, we implement the Quasi-Newton’s method (QNM) to linearize the nonlinear equations. Finally, based on the theory of reproducing kernel method (RKM), a stable numerical scheme is proposed to resolve the linear equations. The reliability and efficiency are verified by numerical experiments.

MSC:

65L03 Numerical methods for functional-differential equations
34A08 Fractional ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abdeljawad, T.; Baleanu, D., Caputo q-fractional initial value problems and a q-analogue Mittag-Leffler function, Commun. Nonlinear Sci. Numer. Simul., 12, 4682-4688 (2011) · Zbl 1231.26006
[2] Annaby, M.; Mansour, Z., Q-Fractional Calculus and Equations (2012), Springer: Springer Heidelberg, New York · Zbl 1267.26001
[3] Jarad, F.; Abdeljawad, T.; Baleanu, D., Stability of q-fractional non-autonomous systems, Nonlinear Anal. RWA, 14, 780-784 (2013) · Zbl 1258.34014
[4] Aral, A.; Gupta, V.; Agarwal, R., Applications of Q-Calculus in Operator Theory (2013), Springer: Springer Heidelberg, New York · Zbl 1273.41001
[5] Salahshour, S.; Ahmadian, A.; Chan, C., Successive approximation method for captuo q-fractional IVPs, Commun. Nonlinear Sci. Numer. Simul., 24, 153-158 (2015) · Zbl 1440.34011
[6] Tang, Y.; Zhang, T., A remark on the q-fractional differential equations, Appl. Math. Comput., 350, 198-208 (2019) · Zbl 1428.34023
[7] Zhang, T.; Tang, Y., A difference method for solving the q-fractional differential equations, Appl. Math. Lett., 98, 292-299 (2019) · Zbl 1468.65086
[8] Liu, W.; Han, W.; Lu, H.; Li, S.; Gao, J., Reproducing kernel element method. Part I: Theoretical formulation, Comput. Method Appl. M., 193, 933-951 (2004) · Zbl 1060.74670
[9] Xu, M.; Lin, Y., Simplified reproducing kernel method for fractional differential equations with delay, Appl. Math. Lett., 52, 156-161 (2016) · Zbl 1330.65102
[10] Griebel, M.; Rieger, C., Reproducing kernel Hilbert spaces for parametric partial differential equations, SIAM-ASA J. Uncertain., 5, 111-137 (2017) · Zbl 1371.35345
[11] Niu, J.; Xu, M.; Lin, Y.; Xue, Q., Numerical solution of nonlinear singular boundary value problems, J. Comput. Appl. Math., 331, 42-51 (2018) · Zbl 1377.65095
[12] Geng, F. Z.; Qian, S. P., Modified reproducing kernel method for singularly perturbed boundary value problems with a delay, Appl. Math. Model., 39, 5592-5597 (2015) · Zbl 1443.65090
[13] Xu, M.; Zhang, L.; Tohidi, E., A fourth-order least-squares based reproducing kernel method for one-dimensional elliptic interface problems, Appl. Numer. Math., 162, 124-136 (2021) · Zbl 1460.65095
[14] Geng, F.; Qian, S., An optimal reproducing kernel method for linear nonlocal boundary value problems, Appl. Math. Lett., 77, 49-56 (2018) · Zbl 1380.65129
[15] Li, X.; Wu, B., A new kernel functions based approach for solving 1-D interface problems, Appl. Math. Comput., 380, Article 125276 pp. (2020) · Zbl 07200822
[16] Geng, F.; Wu, X., Reproducing kernel functions based univariate spline interpolation, Appl. Math. Lett., 122, Article 107525 pp. (2021) · Zbl 1524.65292
[17] Cui, M.; Lin, Y., Nonlinear numerical analysis in the reproducing kernel space, Nova Sci. (2009) · Zbl 1165.65300
[18] Zhang, T.; Guo, Q., The solution theory of the nonlinear q-fractional differential equations, Appl. Math. Lett., 104, Article 106282 pp. (2020) · Zbl 1439.39004
[19] Tang, Y.; Zhang, T., A remark on the q-fractional order differential equations, App. Math. Comput., 350, 198-208 (2019) · Zbl 1428.34023
[20] Xu, M.; Niu, J.; Lin, Y., An efficient method for fractional nonlinear differential equations by quasi-Newton’s method and simplified reproducing kernelmethod, Math. Methods Appl. Sci., 41, 5-14 (2018) · Zbl 1387.65068
[21] Zhao, Z.; Lin, Y.; Niu, J., Convergence order of the reproducing kernel method for solving boundary value problems, Math. Model. Anal., 21, 466-477 (2016) · Zbl 1488.34339
[22] Li, X.; Wu, B., Error estimation for the reproducing kernel method to solve linear boundary value problems, J. Comput. Appl. Math., 243, 10-15 (2013) · Zbl 1261.65085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.